Extremal inclusions in nonlinear conductivity

We consider two-phase composites whose microstructures are two-dimensional and generated by the periodic replication of a convex polygonal cell containing a single inclusion embedded in a matrix. Adopting the framework of nonlinear conductivity, we address the problem of finding the inclusion shape that optimizes the effective energy. A conceptually simple but numerically effective approach is presented, in which the inclusion shape is parameterized by the Fourier coefficients of a scalar periodic function f that defines its polar representation. Truncating the Fourier expansion to a finite order turns the shape optimization problem into a finite-dimensional constrained optimization problem that can be solved using a numerical algorithm of choice. Explicit expressions of the function to optimize and its gradient are provided and can easily be evaluated from a finite-element model. The proposed approach is applied to perfectly conducting inclusions in a power law matrix. Results for the three types of regular tessellations (square, hexagonal and triangular) are presented and compared with the Vidgergauz [23, 24] microstructures giving the extremal inclusions in the linear case. The proposed method gives very simple representations of the extremal inclusions, which could useful for manufacturing the microstructures considered. The obtained nonlinear effective conductivities are compared with known Hashin–Shtrikman-type nonlinear bounds, which contributes to shed some light on the optimality of those bounds.